Over a region of small lateral extent the equilibrium-surface at any time may be regarded as sensibly a plane. But for a land-locked sea the condition that the total volume of water remains constant, requires its surface, whenever plane, to pass through the centroid of its area. It follows that for a small land-locked sea there is no rise and fall of the corrected equilibrium-tide at the centroid of its area.
The acceleration of a particle of water will consist of two parts, one due to the rate of change of the current relative to the earth and the other to the earth's rotation. If we take I/ to denote the angular speed of rotation of the earth then the vertical component at a place of co-latitude 8 is Ocose, and this we shall denote by w. The second part of the accelera tion will be given by multiplying the current by 2w and turning it through a right angle to the left. The Cartesian components of accelera tion take the form— The form of these equations shows that for the oscillatory motion of the tides the current-components u, v will be the same at all depths.
In this argument a number of factors have been left out of account. When frictional forces have to be allowed for, the current-components will not be the same at all depths, and in the equation of continuity v must be interpreted as mean values from sea-floor to sea-surface, while frictional terms must be added to the dynamical equations. The
sea-floor may itself be subject to a small variable elevation, say due to earth-tides, and if we then take r to denote the elevation of the sea surface relative to the floor, the equations (I), (I • 1) will stand, but in the formulae for pressure-gradients must be replaced by rd-re. The dis placements of water and solid earth will themselves create a disturbance in the gravitational forces, and on this account a term must be added to Harmonic Motions.—(9). Suppose now that = cos n (t — TO) as in a single harmonic constituent, ro being functions of 0, 4 but not of t. Then the solution of the equations (I), (2) and (3) of §8 will be of the form =Hocos n(t— r), LI cos n(1 — , v= V cos n(t— 2) , where H, U, V, r, 71,72 are all functions of 0, ct. but are independent of t. These expressions represent a harmonic constituent of the tides themselves as distinct from one of the equilibrium-form. At any place the elevation and current-components vary harmonically with the same speed as the equilibrium-form. The functions II and 7 will in general be quite differ ent from Ho and ro, so that the actual tidal elevation will be quite differ ent from the equilibrium-elevation. If = r2 for any place, the current at the place will flow in one line only, but in general and T2 will not be the same and consequently the currents will be rotatory. Any number of independent solutions of the equations (I), (2) and (3) may be super posed by simple addition, and it follows that for any constituent in the equilibrium-form there will be a constituent of equal speed in the actual tides. These constituents are known by the symbols and names already given for the equilibrium-form, and by methods of analysis (§2r) they may be disentangled from the observations of elevations and currents taken at any one place. The dynamical theory here sketched forms the basis of the harmonic analysis of tidal observations, a process which has been carried out for the elevations at over a thousand stations scattered round the coasts of all the oceans and seas, and for the elevations and currents at a few stations away from the shore in certain shallow seas.